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Κανονικοποίηση
Κανονικοποίησις normalization thumb|300px| [[Κανονικοποίηση ]] thumb|300px| [[Κανονικοποίηση ]] thumb|300px| [[Κανονικοποίηση ]] thumb|300px| [[Κανονικοποίηση ]] - Μία διαδικασία. Ετυμολογία Η ονομασία "Κανονικοποίηση" σχετίζεται ετυμολογικά με την λέξη "κανόνας". Εισαγωγή For now, consider the simple case of a non-relativistic single particle, without spin, in one spatial dimension. More general cases are discussed below. Position-space wave functions The state of such a particle is completely described by its wavefunction, : \Psi(x,t)\,, where x'' is position and ''t is time. This is a complex-valued function of two real variables''x'' and t''. For one spinless particle in 1d, if the wave function is interpreted as a probability amplitude, the square modulus of the wave function, the positive real number : \left|\Psi(x, t)\right|^2 = {\Psi(x, t)}^{*}\Psi(x, t) = \rho(x, t), is interpreted as the probability density that the particle is at ''x. The asterisk indicates the complex conjugate. If the particle's position is measured, its location cannot be determined from the wave function, but is described by a probability distribution. The probability that its position x'' will be in the interval ''a ≤ x'' ≤ ''b is the integral of the density over this interval: : P_{a\le x\le b} (t) = \int\limits_a^b d x\,|\Psi(x,t)|^2 :where t'' is the time at which the particle was measured. This leads to the '''normalization condition': : \int\limits_{-\infty}^\infty d x \, |\Psi(x,t)|^2 = 1\,, because if the particle is measured, there is 100% probability that it will be somewhere. For a given system, the set of all possible normalizable wave functions (at any given time) forms an abstract mathematical vector space, meaning that it is possible to add together different wave functions, and multiply wave functions by complex numbers (see vector space for details). Technically, because of the normalization condition, wave functions form a projective space rather than an ordinary vector space. This vector space is infinite-dimensional, because there is no finite set of functions which can be added together in various combinations to create every possible function. Also, it is a Hilbert space, because the inner product of two wave functions Ψ1 and Ψ2 can be defined as the complex number (at time t'')The functions are here assumed to be elements of ''L''2, the space of square integrable functions. The elements of this space are more precisely equivalence classes of square integrable functions, two functions declared equivalent if they differ on a set of Lebesgue measure0. This is necessary to obtain an inner product (that is, (Ψ, Ψ) = 0 ⇒ Ψ ≡ 0) as opposed to a '''semi-inner product'. The integral is taken to be the Lebesque integral. This is essential for completeness of the space, thus yielding a complete inner product space = Hilbert space. : ( \Psi_1 , \Psi_2 ) = \int\limits_{-\infty}^\infty d x \, \Psi_1^*(x, t)\Psi_2(x, t). More details are given below. Although the inner product of two wave functions is a complex number, the inner product of a wave function with itself, : (\Psi,\Psi) = \|\Psi\|^2 \,, is always a positive real number. The number!!Ψ!! (not !!Ψ!!2}}) is called the norm of the wave function , and is not the same as the modulus Ψ }}. If (Ψ, Ψ) = 1, then Ψ is normalized. If Ψis not normalized, then dividing by its norm gives the normalized function Ψ }}. Two wave functions and are orthogonal if 0}}. If they are normalized and orthogonal, they are orthonormal. Orthogonality (hence also orthonormality) of wave functions is not a necessary condition wave functions must satisfy, but is instructive to consider since this guarantees linear independence of the functions. In a linear combination of orthogonal wave functions we have, : \Psi = \sum_n a_n \Psi_n \,,\quad a_n = \frac{( \Psi_n , \Psi )}{( \Psi_n , \Psi_n )} If the wave functions were nonorthogonal, the coefficients would be less simple to obtain. In the Copenhagen interpretation, the modulus squared of the inner product (a complex number) gives a real number : \left|(\Psi_1,\Psi_2)\right|^2 = P\left(\Psi_2 \rightarrow \Psi_1\right) \,, which, assuming both wave functions are normalized, is interpreted as the probability of the wave function Ψ2 "collapsing" to the new wave function Ψ1 upon measurement of an observable, whose eigenvalues are the possible results of the measurement, with Ψ1}} being an eigenvector of the resulting eigenvalue. This is the Born rule, and is one of the fundamental postulates of quantum mechanics. At a particular instant of time, all values of the wave function Ψ(x'', ''t) are components of a vector. There are uncountably infinitely many of them and integration is used in place of summation. In Bra–ket notation, this vector is written : |\Psi(t)\rangle = \int dx \Psi(x,t) |x\rangle and is referred to as a "quantum state vector", or simply "quantum state".There are several advantages to understanding wave functions as representing elements of an abstract vector space: *All the powerful tools of linear algebra can be used to manipulate and understand wave functions. For example: **Linear algebra explains how a vector space can be given a basis, and then any vector in the vector space can be expressed in this basis. This explains the relationship between a wave function in position space and a wave function in momentum space, and suggests that there are other possibilities too. **Bra–ket notation can be used to manipulate wave functions. *The idea that quantum states are vectors in an abstract vector space is completely general in all aspects of quantum mechanics and quantum field theory, whereas the idea that quantum states are complex-valued "wave" functions of space is only true in certain situations. The time parameter is often suppressed, and will be in the following. The coordinate is a continuous index. The are the basis vectors, which are orthonormal so their inner product is a delta function; : \langle x' | x \rangle = \delta(x' - x) thus : \langle x' |\Psi\rangle = \int dx \Psi(x) \langle x'|x\rangle = \Psi(x') and : |\Psi\rangle = \int dx |x\rangle \langle x |\Psi\rangle = \left( \int dx |x\rangle \langle x |\right) |\Psi\rangle which illuminates the identity operator : I = \int dx |x\rangle \langle x | \,. Finding the identity operator in a basis allows the abstract state to be expressed explicitly in a basis, and more (the inner product between two state vectors, and other operators for observables, can be expressed in the basis). Momentum-space wave functions The particle also has a wave function in momentum space: : \Phi(p,t) where is the momentum in one dimension, which can be any value from to , and is time. Analogous to the position case, the inner product of two wave functions and can be defined as: : (\Phi_1 , \Phi_2 ) = \int\limits_{-\infty}^\infty d p \, \Phi_1^*(p, t)\Phi_2(p, t) \,. One particular solution to the time-independent Schrödinger equation is : \Psi_p(x) = e^{ipx/\hbar}, a plane wave, which can be used in the description of a particle with momentum exactly , since it is an eigenfunction of the momentum operator. These functions are not normalizable to unity (they aren't square-integrable), so they are not really elements of physical Hilbert space. The set : \{\Psi_p(x, t), -\infty \le p \le \infty\} forms what is called the momentum basis. This "basis" is not a basis in the usual mathematical sense. For one thing, since the functions aren't normalizable, they are instead normalized to a delta function, : (\Psi_{p},\Psi_{p'}) = \delta(p - p'). For another thing, though they are linearly independent, there are too many of them (they form an uncountable set) for a basis for physical Hilbert space. They can still be used to express all functions in it using Fourier transforms as described next. Relations between position and momentum representations The and representations are : \begin{align} |\Psi\rangle = I|\Psi\rangle &= \int |x\rangle \langle x|\Psi\rangle dx = \int \Psi(x) |x\rangle dx,\\ |\Psi\rangle = I|\Psi\rangle &= \int |p\rangle \langle p|\Psi\rangle dp = \int \Phi(p) |p\rangle dp.\end{align} Now take the projection of the state onto eigenfunctions of momentum using the last expression in the two equations, : \int \Psi(x) \langle p|x\rangle dx = \int \Phi(p') \langle p|p'\rangle dp' = \int \Phi(p') \delta(p-p') dp' = \Phi(p). Then utilizing the known expression for suitably normalized eigenstates of momentum in the position representation solutions of the free Schrödinger equation : \langle x | p \rangle = p(x) = \frac{1}{\sqrt{2\pi\hbar}}e^{\frac{i}{\hbar}px} \Rightarrow \langle p | x \rangle = \frac{1}{\sqrt{2\pi\hbar}}e^{-\frac{i}{\hbar}px}, one obtains : \Phi(p) = \frac{1}{\sqrt{2\pi\hbar}}\int \Psi(x)e^{-\frac{i}{\hbar}px}dx\,. Likewise, using eigenfunctions of position, : \Psi(x) = \frac{1}{\sqrt{2\pi\hbar}}\int \Phi(p)e^{\frac{i}{\hbar}px}dp\,. The position-space and momentum-space wave functions are thus found to be Fourier transforms of each other. The two wave functions contain the same information, and either one alone is sufficient to calculate any property of the particle. As representatives of elements of abstract physical Hilbert space, whose elements are the possible states of the system under consideration, they represent the same state vector, hence identical physical states, but they are not generally equal when viewed as square-integrable functions. In practice, the position-space wave function is used much more often than the momentum-space wave function. The potential entering the relevant equation (Schrödinger, Dirac, etc.) determines in which basis the description is easiest. For the harmonic oscillator, x'' and ''p enter symmetrically, so there it doesn't matter which description one uses. The same equation (modulo constants) results. From this follows, with a little bit of afterthought, a factoid: The solutions to the wave equation of the harmonic oscillator are eigenfunctions of the Fourier transform in .The Fourier transform viewed as a unitary operator on the space has eigenvalues . The eigenvectors are "Hermite functions", i.e. Hermite polynomials multiplied by a Gaussian function. See for a description of the Fourier transform as a unitary transformation. For eigenvalues and eigenvalues, refer to Problem 27 Ch. 9. Υποσημειώσεις Εσωτερική Αρθρογραφία * κυματοσυνάρτηση *διακριτοποίηση *Γραμμικοποίηση *Ομαλοποίηση *Επανακανονικοποίηση, ορθοκανονικοποίηση *Κανονικότητα *Κβαντική Φυσική * Όριο, κανονικοποίηση Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *Normalize-a-Vector *[ ] Category:Κβαντική Φυσική Κατηγορία:Διαδικασίες